For the state estimation of nonlinear non-Gaussian discrete dynamic systems, based on the Gaussian sum recursive relations, a Gaussian sum squared-root cubature Kalman filter (GSSRCKF) for state estimation is proposed. On the assumption that the probability density functions of process noises, measurement noises and initial condition are denoted by a Gaussian sum or approximated by a Gaussian sum, a bank of squared-root cubature Kalman filters (SRCKF) are used as the Gaussian sub-filters to estimate the state of the system respectively in GSSRCKF. Then, the different filtering results are combined to the global state estimation according to the corresponding weights, which are set as adaptive process parameters at each filtering time. And the effective reduction method is adopted to reduce the computational complexity. The simulation results verify the superiority of the proposed method on filter consistency.